Experimental Investigations of Rise Behavior of Monodispersed

Ind. Eng. Chem. Res. 2010, 49, 10615–10626

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Experimental Investigations of Rise Behavior of Monodispersed/Polydispersed Bubbly Flows in Quiescent Liquids Swapna S. Rabha and Vivek V. Buwa* Department of Chemical Engineering, Indian Institute of Technology-Delhi, New Delhi 110 016, India

The predictive capabilities of continuum CFD models to simulate large-scale dispersed gas-liquid flows depend on the closures used to estimate the interphase coupling forces. The present manuscript shows that different corrections that are applied to correct the drag force for multiple bubble systems lead to different predictions as the gas volume fraction is increased. In the present work, experimental investigations of monodispersed and polydispersed bubbles of different diameters (1.2 e dB e 7.5 mm) rising in quiescent water (0.19 e Eo e 8.72; log Mo ) -10.5) at different gas volume fractions (0.01 < RG < 0.2) are reported. The bubble rise velocity of a single isolated bubble, a bubble rising in a single chain and bubbles rising in multiple chains were compared. The effect of bubble diameter and gas volume fraction on the fluctuations in bubble rise velocities of individual bubbles rising in multiple chains was also investigated. The rise velocities of monodispersed bubble swarms were found to increase with the increase in dB and RG. The number- and time-averaged bubble rise velocity and drag coefficient for monodispersed bubble swarms were investigated as a function of RG. The drag coefficients based on the slip velocity of the bubble swarms for RG < 0.1 were found to decrease with increase in RG and showed good agreement with the previous literature; but for RG > 0.1, the drag coefficient were found to be independent of RG. Further, the rise behavior of poly dispersed bubbles was also investigated. 1. Introduction Several engineering processes involve dispersed gas-liquid flows, for example, in chemical, oil and gas, biochemical industries. Among several different contactors/reactors used for processes involving gas-liquid flows, bubble columns and stirred vessels are used most widely. Over the last two decades, significant research efforts were made to develop computational fluid dynamics (CFD) models based on continuum (Euler/Euler) and discrete particle (Euler/Lagrange) approaches for detailed simulations of unsteady dispersed gas-liquid flows in the abovementioned gas-liquid reactors/contactors. In these approaches, the interphase momentum exchange between the two phases is accounted through various interfacial forces, for example, drag, lift, and virtual mass forces. Several researchers1-10 could predict the dynamic and time-averaged characteristics of the dispersed gas-liquid flows at low gas velocities (VG < 5 cm/s) using the closure laws developed for estimation of drag coefficient acting on a single bubble.11-18 However, for the predictions of dynamic and time-averaged properties of dispersed gas-liquid flows at higher superficial gas velocities (VG > 10 cm/s), different empirical corrections were applied, to the correlations used to estimate drag coefficient for a single bubble, to account for the effect of presence of the neighboring bubbles. Rampure et al.,19 Olmos et al.,20 Kaushik and Buwa,21 Simonnet et al.22 simulated unsteady dispersed gas-liquid flows at higher superficial gas velocities (10 < VG < 40 cm/s) using the drag correlation proposed by Ishii and Zuber16 with a correction factor (for example,CD ) CDo(1 RG)P). The parameter “p” was empirically tuned to higher values (3 and 4) as the gas velocity was increased. This indicates that continuum models, based on closure laws that were developed for single isolated bubbles, can predict the dynamic and timeaveraged characteristics of gas-liquid flows quantitatively for * To whom all correspondence should be addressed. Tel: +91 11 2659 1027. Fax: +91 11 2658 1120. E-mail: vvbuwa@ chemical.iitd.ac.in.

simple systems with low gas volume fractions (VG < 10 cm/s), but often fail to predict the dynamic and time- averaged characteristics of dispersed gas-liquid flows at higher gas volume fractions (VG > 10 cm/s) quantitatively. One of the important reasons for this appear to be the lack of adequate closure models that can account for the effect of bubble shape/ size on different forces (like drag, lift and virtual mass forces) acting on bubbles and more importantly the influence of neighboring bubbles (or volume fraction) on the magnitude of these interphase coupling forces. The present work is focused on experimental investigations of rise behavior of monodispersed and polydispersed bubbles in a (initially) quiescent liquid. The objective of the present work is to quantify the effect of the gas volume fraction (or bubble-bubble interactions) on the mean and the fluctuating bubble rise velocity and finally on the drag coefficient for monodispersed and polydispersed bubbles of different sizes. In the following section, we bring out the present state-of-art of the investigations on the drag correlations developed for single/ multibubble systems and of the dynamics of microscopic bubbly flows that are relevant to the present study. 2. Present State of Art Over the years, several correlations were developed to estimate the drag coefficient (CD). Most of these correlations were deduced either from experimental or from analytical investigations of rise of a single isolated bubble and they account for the effects of the bubble shape and size (dB) and the slip velocity on CD. The important correlations for CD of a single bubble are summarized in Table 1. All these correlations describe CD as a function of the bubble Reynolds number (ReB), except the correlations of Ishii and Zuber16 and Clift et al.12 which describe CD as a function of the Eo¨tvo¨s number (Eo). For ReB < 10, the CD predicted from the above-mentioned correlations decreases linearly with Re (see Figure 1). However, for ReB > 10, some correlations predict CD to be independent

10.1021/ie1006454  2010 American Chemical Society Published on Web 10/07/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 21, 2010

Table 1. Summary of Correlations Available for Estimation of Drag Coefficient (CD) for a Single Bubble Rise investigators Schiller and Naumann15 CD )

{

correlation

0.44

(

)

Dalla Villle28

CD ) 0.63 +

Morsi and Alexander29

B A CD ) Re + + C ReB2 B Re 0-0.1 0.1-1 0.95 ReB e 1800


Figure 1. Comparison of CD as a function of ReB, estimated using various correlations available in the literature (see Table 1).

Figure 2. Comparison of CD as a function of Eo using various correlations available in the literature (see Table 1).

of ReB, whereas others predict a continuous decrease in CD with increase in ReB. The correlations of CD expressed as a function of ReB only account for bubble size (dB) and not the bubble shape. The effect of bubble shape (accounted through Eo) on CD was studied by Ishii and Zuber,16 Clift et al.12 and Tomiyama et al.18 and is shown in Figure 2. For small values of Eo (Eo < 1) i.e. for spherical bubble regime, the CD predicted by all these three correlations increases marginally with the increase in Eo. However, for Eo > 1 (ellipsoidal and wobbling bubble regimes), the correlation of Clift et al.12 shows that the CD is almost independent of Eo, whereas the correlation of Ishii and Zuber16 predicts a sharp increase in CD with increase in Eo. The correlations used to estimate CD for multibubble systems are listed in Table 2. CD for multibubble systems was found to be different from that of a single isolated bubble. Tomiyama et al.17,18 investigated experimentally the rise of bubbles for 0.083 e ReB e 200; 0.13 e Eo e 30 and derived a correlation for CD for bubbles rising in an uncontaminated clean liquid. Their correlation is valid only for systems with viscosity ratio less than 0.3. Ishii and Zuber16 formulated CD as a function of phase volume fraction (RG). Recently, Rusche and Issa23 formulated a correlation to determine CD for dispersed gas-liquid flows at higher RG. A correction function f(RG) was defined in the correlation to account the presence of neighboring bubbles in multiple bubble systems. Of the correlations listed in Table 2, the correlations proposed by Sankaranarayanan et al.24 was

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developed by using microscopic simulations (performed using the lattice-Boltzmann method) of single bubble rise behavior in a periodic box corresponding to different volume fractions. Since, in all of their simulations, Sankaranarayanan et al.24 considered the rise of a single bubble in a periodic box, their CD correlation was applicable only to a regular array of uniformly dispersed bubbles in a liquid and for RG < 0.2. The effect of RG on CD/CD0 predicted using the above-mentioned correlations is shown in Figure 3. The corrections proposed by Ishii and Zuber,16 Tomiyama et al.,18 and Sankaranarayanan et al.24 (n ) -7) showed a decrease in CD/CD0 with increase in RG where as the corrections proposed by Rusche and Issa23 and Sankaranarayanan et al.24 (n ) 4) showed an increase in CD/ CD0 with increase in RG. There exists a large variation in the predicted trends of CD/CD0 as a function of RG (see Figure 3). Besides the studies on correlations for CD, a few experimental studies were reported in the literature that examined the dynamics of microscopic bubbly flows. However, most of these studies considered a uniform bubble size. Zenit et al.25 investigated the effects of RG (0 < RG < 0.2) for dB ) 1.35 mm (3.5% (Eo ) 0.25 and log Mo ) -10.5) on bubble rise velocities for nitrogen-water system. They used dual impedance probe to measure RG, mean and fluctuating bubble rise velocities and mean bubble size. The mean bubble velocity of the bubble swarm was found to decrease with decrease in RG. The variance j B)2) was found to increase of bubble rise velocity (V/B2 ) (VB - V sharply with increase in RG for RG e 0.01 and rather moderately for RG e 0.1. Martinez-Mercado et al.26 studied the rise behavior of monodispersed bubbly flow (dB ) 1.2 - 1.5 mm) in air-water and air-water + glycerol system for 10 < ReB < 500; 0.19 < Eo < 0.29 and -2.5 < log Mo < - 0.49 using the dual impedance probe and found the mean rise velocity of bubbles to decrease with increase in RG. For a constant RG, they found that the mean bubble rise velocity (〈VB〉 ) VB(1 - RG)2.796) of the homogeneous dispersion to decrease with the increase in the liquid viscosity. Initially, the fluctuations in bubble rise velocities were found to increase with increase in RG, but reached a constant value for RG > 0.02. Risso and Ellingsen27 investigated bubble rise velocity fluctuations for a homogeneous bubble swarm (dB ) 2.5 mm, Eo ) 0.21, log Mo ) -10.5) for very small RG (0.005 < RG < 0.01) using double optical fiber probe. Since the size of bubbles was small, the bubble rise behavior was found to be weakly influenced by the hydrodynamic interactions among the bubbles and the mean rise velocity (〈VB〉 ) 1.21VB) of the bubble swarm was found to be close to that of a single bubble. It should also be noted that these experimental observations of mean and fluctuating bubble rise velocities were applicable to monodispersed bubbly flow with dB < 2.5 mm. In addition to the mean bubble rise velocity and its fluctuations, liquid velocity fluctuations are also important to characterize the dynamics of rise behavior of multibubble systems. There are only a few reports available in the literature on the measurements of the liquid velocity fluctuations for multibubble systems. Ze´nit et al.25 performed experimental investigations of rise behavior of monodispersed bubbles using a hot film anemometer to investigate the effect of RG on liquid velocity variance. They observed the liquid velocity variance to increase with increase in RG. Martinez-Mercado et al.26 found the fluctuations in the liquid velocity to increase with the increase in RG, and to become constant for RG > 0.02. Risso and Ellingsen27 also investigated the liquid velocity fluctuations for

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Table 2. Summary of Correlations Available for Estimation of Drag Coefficient (CD) for Multiple Bubble Rise investigators

correlation 35

Richardson and Zaki

Ishii and Zuber16

ReB < 1000

CD ) 0.44RG

RG > 0.8

CD ) CDo(1 - RG)2

Tomiyama et al.17,18 CD ) max

Rusche and Issa23

range of applicability

-2.65

[(

24 (1 + 0.15ReB0.687), ReB

)

9/7 Eo 8 1 + 17.67RG 3 Eo + 4 18.67RG3/2

CD ) CDOf(RG);f(RG) ) exp(K1RG) + RGK2 (K1 ) 3.64 and K2 ) 0.864)

Sankaranarayanan et al.24

]

( )(

4 ∆F 3 FL

CD )

gd3 υ ReRReB 2

)(

1 (1 - RG)n-1

)

where “n’” is the Richardson-Zaki exponent n)

{

( 1.3χ ) χ 3.3 - 51log( ) 1.3

3.3 - 1.7log

χ < 1.3

ReB ) 0.2 - 1000 0.13 e Eo < 30 -10.0 < logMo < 2 0.083 e ReB< 200

ReB g 1000

0.38 e Eo e 9.1 -9.4 < log Mo < -4 100 < ReB < 400 RG < 0.2

χ g 1.3

χ ) ReBMo0.25Eo-0.5

a homogeneous swarm of bubbles (dB ) 2.5 mm) for very small values of RG (0.005 < RG < 0.01) using a dual tip optical fiber probe. From the previous literature on the investigations of microscopic bubbly flows, it can be noted that bubble size considered was very small (dB < 2.5 mm) and fall in the spherical bubble regime of the bubble diagram of Clift et al.12 In most of the previous work, homogeneous (monodispersed) bubbly flow was considered. The rise behavior of monodispersed and poly dispersed bubbles in the wobbling and spherical cap regimes which is more relevant to large-scale gas-liquid flows is not investigated. In this paper, we report the experimental investigations of rise behavior of monodispersed and polydispersed bubbles for air-water system. The effect of RG (0.01 < RG < 0.2) on mean bubble rise velocity and its fluctuations for monodispersed and polydispersed bubbles (1.2 e dB e 7.5 mm)

Figure 3. Effect of RG on CD/CDo predicted using various correlations available in the literature (ReB ) 0.083 - 1000; 0.13 < Eo < 30) (see Table 2).

for air-water system (0.19 e Eo e 8.72 and log Mo ) -10.5) is investigated in detail. A detailed description of the experimental setup and benchmarking of the measurement technique used in the present work is provided in Section 3 and the results are discussed in Section 4. 3. Experimental Setup The schematic of the experimental setup used in the present work is shown in Figure 4. It consists of a rectangular glass column of 600 mm height with a square cross-section of 150 mm width. The column was filled with demineralized water up to a height of 450 mm, and air was introduced at the bottom of the column through an array of stainless steel capillaries of different sizes (100 mm long, 0.26-10 mm inside diameter) to generate bubbles of different size. Experiments were carried out with bubble dispersions with average (sphere-equivalent) bubble diameters of 1.2 ((0.05), 2.9 ((0.1), 4.85 ((0.1), and 7.5 ((0.1) mm corresponding to the gas volume fraction of 0.01 < RG < 0.2. The characteristic dimensionless number for this range of parameters are 120 e ReB e 1821; 0.19 e Eo e 8.72; log Mo ) -10.5 which correspond to the spherical, ellipsoidal, and wobbling bubble regimes. The rise behavior of single isolated bubbles/bubble chains was recorded by using a high speed camera (Fastec Imaging, USA) which is capable of capturing 500 fps at a resolution of 1.1 megapixels. Typical experimental images of the bubble dispersions (dB ) 1.5 ( 0.2 mm at RG ) 0.03, dB ) 3.3 ( 0.1 mm at RG ) 0.09 and dB ) 4.75 ( 0.5 mm at RG ) 0.16) are shown in Figure 5. The equivalent bubble 3 2 diameter (dB) was calculated as dB ) d maxdmin, where dmax is the major axis and dmin is the minor axis of the oblate bubble. The image processing software (Image J) was used to measure the dmax and dmin of bubbles, and the average equivalent bubble diameter was calculated by taking an average over 20 bubbles just after their detachment from the needles. In the present work, the experiments were performed with a 2-D array of bubbles (as shown in Figure 5) and therefore the


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Figure 4. Schematic of the experimental setup used in the present work.

Figure 5. Typical experimental images of the monodispersed bubbly flow (a) dB ) 1.5 ( 0.2 mm at RG ) 0.03, (b) dB ) 3.3 ( 0.2 mm at RG ) 0.09, (c) dB ) 4.75 ( 0.5 mm at RG ) 0.16.

area fraction of bubbles in the interrogation window was considered to be a realistic estimate of the gas volume fraction. In case of 3-D array of bubbles generated by rows of needles arranged one behind the other, it would have been appropriate to consider the actual volume fraction of bubbles. In a 2-D array of bubble, since the interactions from front and rear array of bubbles (along the column depth) are absent, it is appropriate to calculate RG from the area fraction of bubbles. To measure the area fraction, an interrogation area (AT) of different cross sections 50 × 50, 75 × 75, 100 × 100, and 125 × 125 mm2 was considered. The number of bubbles (NB) present in each interrogation area were counted, and the area fraction of the respective interrogation area was calculated as RG ) (πdB2)NB/ AT. The area fraction calculations were repeated for different areas of interrogation window until the area fractions of became

independent of AT. The vertical and horizontal components of rise velocity of individual bubbles were measured by following the trajectories of the respective bubble. Initial experiments were conducted to measure the rise velocity of single air bubbles in quiescent demineralized water for dB ) 1.2 ((0.05), 2.9 ((0.1), 4.85 ((0.1), and 7.5 ((0.1) mm to benchmark the measurements. A single air bubble was released at the center of the column (the distance between the wall and the bubble interface was kept sufficiently large to minimize the wall effects). The rise velocities of bubbles with dB of 1.2, 2.9, 4.85, and 7.5 mm bubble were found to be 10.9, 14.41, 18.76, and 23.2 cm/s, respectively and showed good agreement with the terminal rise velocity measurements of Clift et al.12 for contaminated water. The effect of dimensionless liquid height (H/D) on the bubble rise velocity was studied by performing experiments of single bubble (dB ) 3.973 ( 0.05 mm) rise in a quiescent liquid (water) for three liquid heights 15 cm (H/D ) 1), 30 cm (H/D ) 2), and 45 cm (H/D ) 3). The effect of liquid height on the bubble rise velocity (for dB ) 3.973 ( 0.05 mm) was found to be insignificant (results not shown here). In all further experiments, a liquid height of 45 cm (H/D ) 3) was used. 4. Results and Discussion Experiments were performed with air-water system (0.19 e Eo e 8.72; log Mo ) -10.5) for different bubble size (1.2 ( 0.05 e dB e 7.5 ( 0.5 mm) and gas volume fraction (0.01 < RG < 0.20) and the results are discussed in the following sections. In the first part, the rise behavior of monodispersed bubbly flow is discussed which also include results on effect of neighboring bubbles (or RG) on fluctuating and mean bubble rise velocities and on CD. In the second part, the rise behavior of polydispersed bubbly flow is discussed. 4.1. Monodispersed (Homogeneous) Bubbles Rising in Quiescent Liquid. 4.1.1. Effect of Gas Volume Fraction on Fluctuating Bubble Rise Velocities. The effects of gas volume fraction (RG) and bubble diameter (dB) on fluctuating

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Figure 6. Comparison of rise velocities of isolated single bubbles (dB ) 1.2 ( 0.05 mm), bubbles in a single chain (dB ) 1.2 ( 0.05 mm) and bubbles in multiple chains (dB ) 1.5 ( 0.2 mm, RG ) 0.03).

Figure 7. Comparison of rise velocities of isolated single bubbles (dB ) 2.98 ( 0.1 mm) and bubbles in a single chain (dB ) 2.98 ( 0.1 mm) and bubbles in multiple chains (dB ) 3.3 ( 0.2 mm, RG ) 0.09).

bubble rise velocities of monodispersed bubbles rising in quiescent water (0.19 < Eo < 8.72; log Mo ) -10.5) were studied by performing experiments for dB ) 1.5 ( 0.2, 3.3 ( 0.2, and 4.75 ( 0.5 mm at different gas volume fractions (RG ) 0.015, 0.03 for dB ) 1.5 ( 0.2 mm; RG ) 0.02, 0.06, 0.09 for dB ) 3.3 ( 0.2 mm and RG ) 0.02, 0.05, 0.10, 0.16 for dB ) 4.75 ( 0.5 mm). The comparison of rise velocities of a single isolated bubble, bubbles in a single chain and bubbles in multiple

chains for dB ) 1.5 ( 0.2, 3.3 ( 0.2 mm and 4.75 ( 0.5 mm are shown in Figures 6-8, respectively. In all these figures, the lines corresponding to the rise velocities of single isolated bubbles indicate that measurements were repeated three times. The lines corresponding to the rise velocities of bubbles in a single chain indicate the rise velocities of different bubbles in a single chain considered for the measurements. The lines corresponding to the rise velocities of bubbles in multiple chains


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Figure 8. Comparison of rise velocities of isolated single bubbles (dB ) 4.85 ( 0.1 mm), bubbles in a single chain (dB ) 4.85 ( 0.1 mm) and bubbles in multiple chains (dB ) 4.75 ( 0.5 mm, RG ) 0.16).

Figure 9. Bubble rise velocity distribution for a single isolated bubble (dB ) 1.2 ( 0.05 mm), bubble in a single chain (dB ) 1.2 ( 0.05 mm) and bubbles rising in multiple chains (dB ) 1.5 ( 0.2 mm at RG ) 0.03).

indicate the rise velocities of different bubbles in different chains considered for the measurements. Though, we take only one experimental data set for analysis of the rise behavior of the multiple bubbles, we have also repeated the experiments on the rise behavior of multiple bubbles and the range of fluctuations in the rise velocity remains almost the same for different experiments (results not shown here). Therefore, one experimental set was considered to be adequate for the investigations of rise behavior of multiple bubbles. The effect of gas volume fraction (0.01< RG < 0.2) on the bubble rise velocity distributions for dB ) 1.5 ( 0.2, 3.3 ( 0.2, and 4.75 ( 0.5 mm are shown in Figures 9-11, respectively. The rise velocity of a single isolated bubble (dB ) 1.2 ( 0.05 mm) rising in quiescent water was found to be around 0.11 m/s and the fluctuations in the bubble rise velocity were negligible (see Figure 6). As the number of bubbles increase

(or RG ) 0.03), the rise velocities of individual bubbles in multiple chains were also increased. Since the size of bubble was small, smaller fluctuations in the rise velocities of individual bubbles were observed (see Figure 6). As shown in Figure 9, the bubble rise velocity distributions of individual bubbles rising in multiple chains (dB ) 1.5 ( 0.2 mm) at RG ) 0.03 were found in the range of 0.24-0.42 m/s as compared to the narrow bubble rise velocity distribution observed for a single isolated bubble. As the bubble size was increased to dB ) 3.3 ( 0.2 mm, the rise velocity of an individual bubble rising in a single chain of bubbles was found to be higher than that of a single isolated bubble (VB ) 0.14 m/s) which clearly shows the effect of the leading and trailing bubbles (see Figure 7). Unlike for the single isolated bubbles, significant fluctuations were seen in the rise velocities of the bubbles rising in multiple chains (NB ) 9). The rise behavior of bubbles in multiple chains clearly indicates the effect of the neighboring bubbles (interaction with leading and trailing bubbles and with the bubbles rising along the sides). The bubble rise velocity distributions of individual bubbles (dB ) 3.3 ( 0.2 mm) in multiple chains were found to be wider (VB ) 0.2-0.4 m/s for RG ) 0.02 and VB ) 0.18-0.48 m/s for RG ) 0.09) as compared to that of a single isolated bubble (dB ) 2.98 ( 0.1 mm). It should be noted that the velocity distributions of the bubbles rising in multiple chains at RG ) 0.02 and 0.09 were found to shift toward high velocity regions as shown in Figure 10 (a) and (b). The fluctuations in the rise velocities of the individual bubbles in multiple chains were also found to increase with the increase in RG. For dB ) 4.75 ( 0.5 mm, similar results to that of dB ) 3.3 ( 0.2 mm, were observed for increase in the magnitude of rise velocity and also in the magnitude of fluctuations in the rise velocities with increase in RG to 0.16 (see Figure 8). As compared to the narrow rise velocity distribution (0.20-0.24 m/s) observed for a single isolated bubble (dB ) 4.85 ( 0.1 mm), rise velocity distributions of the individual bubbles in multiple chains were found much

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Figure 10. Bubble rise velocity distribution of a single isolated bubble (dB ) 2.98 ( 0.2 mm), bubble in a single chain (dB ) 2.98 ( 0.2 mm) and bubbles in multiple chains for dB ) 3.3 ( 0.2 mm at (a) RG ) 0.02 and (b) RG ) 0.09.

wider (0.25 - 0.35 m/s for RG ) 0.05, 0.2 - 0.52 for RG ) 0.10 and 0.16) as shown in Figures 11 (a)-(c), respectively. The rise velocity distribution graphs were also found to shift toward higher velocities. This indicates the increase in interactions among the bubbles with increase in RG and also with increase with dB. 4.1.2. Effect of Gas Volume Fraction on Number- And Time-Averaged Bubble Rise Velocity. The number-averaged bubble rise velocities of the bubbles in multiple chains in a homogeneous bubble dispersion for dB ∼ 4.75 ( 0.5 mm for different values of RG are shown in Figure 12. The numberaveraged bubble rise velocity 〈VB〉 was calculated as 〈VB(t)〉 )

1 NB

NB

∑V

B,i(t)

(1)

i)1

The fluctuations in the rise velocity of individual bubbles were found to decrease due to the number- averaging and the 〈VB〉 was found to increase with increase in RG. In order to quantify the effect of RG on 〈VB〉, averaging was also carried out over time to obtain the number- and time- averaged (after discarding j B〉 of the bubble the initial transients) bubble rise velocity 〈V dispersion as:

Figure 11. Bubble rise velocity distribution of a single isolated bubble (dB ) 4.85 ( 0.1 mm) bubble in a single chain (dB ) 4.85 ( 0.1 mm) and bubbles rising in multiple chains (dB ) 4.57 ( 0.5 mm) at (a) RG ) 0.05, (b) RG ) 0.10, and (c) RG ) 0.16.

j B〉 ) 1 〈V T

∫ 〈V (t)〉dt B

(2)

Figure 13 shows the effect of RG on number- and timej B〉) for dB ) 4.75 ( 0.05, 3.3 averaged bubble rise velocity (〈V j B〉 of the homogeneous ( 0.2, and 1.5 ( 0.2 mm. The 〈V dispersion of small bubbles (dB ) 1.5 ( 0.2 mm) at RG ) 0.03 was 0.34 m/s which shows good agreement with the predicted mean bubble rise velocity VB ) 0.33 m/s (Spelt and Sangani36) and experimentally measured mean bubble rise velocity VB ) 0.28 m/s (Zenit et al.25) for dB ) 1.35 ( 0.03 mm at RG ) j B〉 0.03. For large bubbles (dB ∼ 3.3 mm and 4.75 mm), the 〈V of the homogeneous dispersion was found to increase with the increase in RG and remain almost constant for RG > 0.06 for dB ∼ 3.3 mm and RG > 0.10 for dB ∼ 4.75 mm (see Figure 13). At low gas volume fraction (RG < 0.06 for dB ∼ 3.3 mm and RG < 0.10 for dB ∼ 4.75 mm), the bubbles were observed to rise in a regular array and the horizontal distance between the bubbles


Figure 12. Number-averaged bubble rise velocity as a function time at RG ) 0.02. 0.05, 0.10, 0.16 for dB ) 4.75 ( 0.5 mm.

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Figure 14. CD/CD0 based on slip velocity of the bubble swarm as a function of RG for dB ) 4.75 ( 0.5, 3.3 ( 0.2, and 1.5 ( 0.2 mm.

Figure 13. Number- and time-averaged bubble rise velocity as a function of RG for dB ) 4.75 ( 0.5, 3.3 ( 0.2, and 1.5 ( 0.2 mm.

was large. Under such circumstance, the rise behavior of individual bubbles is influenced by the wake of the leading bubble and thus the bubbles experience a higher rise velocity than that of a single isolated bubble. But, due to smaller horizontal distance between bubbles at high gas volume fraction (RG > 0.06 for dB ∼ 3.3 mm and RG > 0.10 for dB ∼ 4.75 mm), the lateral interactions among the bubbles in neighboring chains increase and dominate over the wakes effects induced by the bubbles in an individual bubble chain and diminishes the effect j B〉. The 〈V j B〉 of the homogeneous bubble dispersion RG on the 〈V was further used to calculate the drag force (CD) and the results are discussed in the following section. 4.1.3. Effect of Gas Volume Fraction on Drag Coefj B〉 ficient. The drag coefficient based on slip velocity (VR ) 〈V - VL) of the homogeneous bubble dispersions was calculated from the equation of bubble motion. (FG + 0.5FL)

bR dV 3 CD b b )VR × ∇ × b VL + F |V |V - CLFLb dt 4 dB L R R b (3) (FL - FG)g

Equation 3 was simplified by neglecting lift force, virtual mass force.

Figure 15. A snapshot of poly dispersed bubbles [dB1 ) 7.5 ( 0.1 mm (bubble chain at the center), dB2 and dB3 ) 4.75 ( 0.5 mm (two side chains), dB4 and dB5 ) 3.3 ( 0.2 mm (two extreme side chains)] at RG ) 0.03.

CD )

(

)

1 4 FL - FG gdB 2 3 FL b VR

(4)

However, in the absence of the instantaneous liquid velocity measurements for multiple bubbles rising in a (initially) quiescent liquid at different volume fractions and bubble sizes, we used the simultaneous measurements of gas and liquid velocities reported in the previous literature.39-42 Lindken and Merzkirch39,40 measured the bubble and liquid velocities for multiple bubbles rising in a (initially) quiescent liquid (mean dB ) 5.5 mm, RG ) 0.025) and reported that the mean liquid velocity to be 5 times lower than the gas velocity (VL ) 0.2VG). Similarly, Border and Sommerfeld41 also measured the gas and liquid velocity for multiple bubbles rising in a (initially) quiescent liquid (dB ) 2 - 4 mm; RG ) 0.0075-0.0175) and found that the mean liquid velocity to be VL ) 0.2VG Further, Sathe et al.42 measured the instantaneous bubble rise velocity and liquid velocity for dB ) 0.3-15 mm and RG ) 0.035 in a

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Figure 16. Comparison of rise velocities of single isolated single bubbles (dB ) 2.98, 4.85, and 7.84 mm) and bubbles in multiple chains for a polydispersed system (dB ) 3.3, 4.75, and 7.5 mm, RG ) 0.03).

(initially) quiescent liquid. They found significantly higher liquid velocity (VL ) 0.6VG) as compared to the liquid velocities reported earlier. Following the above reports, we used an approximate VL ) 0.2VG in the calculation of CD. It should be noted that the precise effects of change in RG and dB on VL, and therefore on CD, still remains unaccounted for. Figure 14 shows CD/CDO of the monodispersed bubbles as a function of RG for dB ) 1.5 ( 0.2, 3.3 ( 0.2 and 4.75 ( 0.5 mm. CD0 is the drag coefficient of a single isolated bubble calculated using the measured rise velocity of that bubble. At low RG, CD/CD0 decreases sharply with increase in RG. With further increase in RG, CD/CDO became almost constant and was 0.32 for dB ) 4.75 mm, 0.19 for dB ) 3.3 mm and 0.12 for dB ) 1.5 mm. The experimental observations reported in the present work at low gas volume fraction (RG < 0.1) agree reasonably well with the corrections proposed by Ishii and Zuber,16 Tomiyama et al.18 and Sankaranarayanan et al.24 (n ) -7) which reported a decrease in CD/CD0 with increase RG (see Figure 3). But at higher RG, our experimental observations showed CD/CDO to remain constant with further increase in RG. In most of the previous investigations (see Table 2 and Figure 3), CD/CDO was corrected using some function of RG to account for the effect of neighboring bubbles. It should also be noted from the present investigations that the correction applied to CD/CDO to account for the effect of neighboring bubbles is not only a function of RG, but is also a function of dB. Further experimental investigations are in progress to study the effect of dB (importantly for larger bubbles dB > 5 mm) and the effect of liquid properties such that quantitative correction factors can be proposed. 4.2. Polydispersed (Heterogeneous) Bubbles Rising in Quiescent Liquid. In order to investigate the rise behavior of polydispersed bubble chains, experiments were performed with five chains of three different bubble sizes (dB1 ) 7.5 mm, dB2 and dB3 ) 4.75 mm, dB4 and dB5 ) 3.3 mm) dispersed in quiescent water in an arrangement shown in Figure 15. The

individual bubble rise velocities in polydispersed system at RG ) 0.03 and the rise velocities of single isolated bubbles of corresponding dB are shown in Figure 16. Unlike quasi-steady rise velocities observed for a single isolated bubble with small fluctuations (see Figure 16), the rise velocity of the large bubble dB1 (∼ 7.5 mm) does not attain a quasi-steady state for the liquid height considered in the present work. Since the bubbles dB2 and dB3 () 4.75 mm) were released on each side of dB1 () 7.5 mm), the effect of the wake induced by the large bubble dB1 on the bubbles dB2 and dB3 was more pronounced. As a result, the rise velocities of dB2 and dB3 (individual bubble rising in a chain on each side of dB1) were also not found to reach the quasi-steady state and were almost equal to that of dB1. The bubbles dB4 and dB5 () 3.3 mm) were less affected by dB1 and their number- and time-averaged bubble rise velocities were found to be 0.29 and 0.32 m/s respectively which were close to that of a single bubble rising in a single j B ) 0.32 shown in Figure 7). To quantify the effect of chain (V poly dispersity, an average CD based on slip velocity (VR ) 0.29 m/s) and on average () 5.16 mm) of the bubble dispersion at RG ) 0.03 was calculated and was found to be around 0.77. Further investigations to quantify the effect of poly dispersity on rise behavior of bubbles rising at different volume fractions are in progress and will be reported separately. 5. Conclusions The rise behaviors of both monodispersed/polydispersed bubbles of different diameters (1.2 e dB e 7.5 mm) rising in quiescent water (0.19 e Eo e 8.72; log Mo ) -10.5) at different gas volume fractions (0.01 < RG < 0.2) were experimentally investigated. The effect of neighboring bubbles on the bubble rise velocities of the individual bubbles rising in multiple chains at different gas volume fractions for different bubble diameters was investigated. Further, the CD based on slip velocity of the bubble dispersion for different dB and RG was


investigated. The effect of neighboring bubbles on the bubble rise velocity and on CD of the individual bubbles rising in a polydispersed system was investigated. The key conclusions of the present work are as follows. For monodispersed bubbles rising in quiescent liquid, the bubble rise velocity of the individual bubbles rising in multiple chains was found to be higher than that of bubbles rising in a single chain and single isolated bubble. The fluctuations in bubble rise velocities of individual bubbles rising in monodispersed bubbly flows were found to increase with increase in j B〉 of the monodispersed RG and also with increase in dB. The 〈V bubbles was found to increase with increase in RG and remain almost constant for RG > 0.06 for dB ∼ 3.3 mm and RG > 0.10 j R of the monodispersed for dB ∼ 4.75 mm. The CD based on V bubble swarm at low RG was found to decrease with increase in RG and agreed well with the literature reports.16,18,24 But at RG > 0.1, CD was found to be independent of RG. The CD/CD0 of a monodispersed system was found to be a function of RG and dB. Unlike monodispersed bubbly flows (for bubbles with dB < 5 mm), rise velocities of bubble in a polydispersed system did not attain a quasi-steady state. Further work is in progress on systematic investigations of rise behavior of monodispersed and polydispersed bubbly flows to study the effect of bubble diameter (dB > 5 mm) and RG on the CD. Such information will facilitate the development of closures that can account for dB and RG on the magnitude of drag force. Acknowledgment S.S.R. is grateful to the Ministry of Human Resource Development (MHRD), India for providing the research fellowship. Appendix Notations AT ) area of the interrogation window, mm2 CD ) drag coefficient, CDo ) drag coefficient of a single isolated bubble, CL ) lift coefficient, D ) column depth, mm dB ) sphere equivalent bubble diameter, mm Eo ) Eo¨tvo¨s number () g∆dB2/σ), g ) gravitational constant, m s-2 H ) column height, mm Mo ) Morton number () g∆ FL4/FL2σ3), NB ) number of bubbles, n ) Richardson-Zaki exponent, ReB ) Reynold number () VBFLdB/µL), So ) density ratio, t ) time, s VB ) bubble rise velocity, ms-1 〈VB〉 ) number-averaged bubble rise velocity, ms-1 j B〉 ) number- and time-averaged bubble rise velocity, ms-1 〈V We ) Weber number () VB2FLdB/σ), VL ) liquid velocity, ms-1 VR ) relative or slip velocity () VG - VL), ms-1 Greek Letters R ) volume fraction, µ ) molecular viscosity, Pa · s F ) density, kgm-3 σ ) surface tension, Nm-1

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Subscripts and Superscripts G ) gas L ) liquid B ) bubble

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ReceiVed for reView March 16, 2010 ReVised manuscript receiVed August 9, 2010 Accepted August 25, 2010 IE1006454

Experimental Investigations of Rise Behavior of Monodispersed

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